2013-41.md

course	course_year	question_number	tags	title	year
Linear Algebra	IB	41	IB 2013 Linear Algebra	Paper 3, Section II, E	2013

Let $V$ and $W$ be finite dimensional real vector spaces and let $T: V \rightarrow W$ be a linear map. Define the dual space $V^{}$ and the dual map $T^{}$. Show that there is an isomorphism $\iota: V \rightarrow\left(V^{}\right)^{}$ which is canonical, in the sense that $\iota \circ S=\left(S^{}\right)^{} \circ \iota$ for any automorphism $S$ of $V$.

Now let $W$ be an inner product space. Use the inner product to show that there is an injective map from im $T$ to $\operatorname{im} T^{*}$. Deduce that the row rank of a matrix is equal to its column rank.